A Semicircular Wire Pqs Of Radius R

You're staring at a diagram. Practically speaking, points P and S mark the ends, Q sits at the midpoint of the arc. Maybe it carries a uniform charge Q. Radius r. A thin wire, bent into a perfect half-circle. Maybe there's a current I flowing through it. Maybe you're just trying to find its center of mass.

Whatever the setup, the semicircular wire is one of those classic physics problems that shows up everywhere — introductory mechanics, electromagnetism, even advanced topics like antenna theory. And every time, students make the same handful of mistakes.

Let's walk through it properly. No hand-waving. That's why no "it can be shown that. " Just the geometry, the calculus, and the places where intuition fails.

What Is a Semicircular Wire PQS of Radius r?

Picture a wire of negligible thickness, uniform linear density (mass per unit length λ, or charge per unit length λ, or just carrying current I). Now, it's bent into a half-circle of radius r. On the flip side, the straight-line distance between the endpoints P and S is 2r — that's the diameter. The curved length is πr.

The label "PQS" just names three points: P and S at the ends of the diameter, Q at the top of the arc (the midpoint). Sometimes textbooks use this notation to define orientation: PQS goes counterclockwise, or the current flows P → Q → S. The letters themselves don't change the physics — they're just reference points.

What makes this geometry useful? A semicircle has reflection* symmetry across the vertical axis (the line through Q and the center O). Practically speaking, a full circle has rotational symmetry about its center. Symmetry. That single axis of symmetry simplifies almost every calculation you'll do with it.

Why This Geometry Shows Up Everywhere

You'll meet the semicircular wire in three main contexts:

Center of mass problems — usually the first time you see it in mechanics. A uniform wire bent into a semicircle. Where's the center of mass? It's not at the center of the full circle (that's empty space). It's not at Q. It's somewhere on the symmetry axis, below Q. Finding it teaches you how to set up line integrals with symmetry.

Electric field of a charged semicircle — a standard E&M problem. Uniform line charge λ on the wire. Find the field at the center O, or at some point on the axis. The symmetry kills the horizontal components. Only vertical survives. You integrate dE sinθ or dE cosθ depending on your coordinate choice.

Magnetic field at the center from a current — Biot-Savart law. Current I flows P → Q → S. The field at O points perpendicular to the plane (into or out of the page by right-hand rule). Magnitude is half that of a full loop: B = μ₀I/4r. Clean result. Shows up on exams constantly.

There are others — moment of inertia about different axes, inductance of a semicircular loop, radiation pattern of a semicircular antenna — but those three are the pillars.

How to Set Up the Calculus (Without Getting Lost)

Every calculation on this geometry follows the same pattern. Master the setup once, and you can adapt it to mass, charge, current, whatever.

Choose your coordinate system wisely

Place the center of the full circle at the origin. Day to day, the arc PQS sits in the upper half-plane (y ≥ 0). Let the diameter PS lie on the x-axis, with P at (-r, 0) and S at (r, 0). The midpoint Q is at (0, r).

Now parameterize the wire by angle θ, measured from the +x axis. A point on the wire has coordinates:

• x = r cos θ
• y = r sin θ

where θ runs from 0 to π (or -π to 0, depending on orientation — pick one and stick with it).

The differential arc length is ds = r dθ*. This is the key. Every line integral becomes an integral over θ with r dθ* as the measure.

The symmetry argument — use it before you integrate

Before writing a single integral, ask: what cancels?

• Horizontal components (x-direction): For every element at angle θ, there's a symmetric element at π - θ with equal magnitude but opposite x-component. They cancel exactly*. Net horizontal = 0. Always. For center of mass, electric field, magnetic field — anything where the source is symmetric.

• Vertical components (y-direction): These add. Both symmetric elements point the same way vertically. So you only need to compute the y-contribution and double it (or integrate 0 to π with the correct trig function).

This one observation saves you from setting up two integrals and wondering why your x-integral gives zero.

For more on this topic, read our article on drag each label to the location of each structure described or check out which of the following is not a facial bone.

The generic integral template

Say you're computing some vector quantity dF from each element. The vertical component is dF_y = dF sin θ (if θ is from +x axis) or dF cos θ (if θ is from +y axis). Pick a convention and be consistent.

The total vertical component: F_y = ∫ dF_y = ∫ (dF sin θ) from θ = 0 to π

where dF depends on the problem:

• Center of mass: dF → dm = λ ds = λ r dθ, and you're finding y_cm = (1/M) ∫ y dm*
• Electric field: dF → dE = k dq* / r² = k λ r dθ* / r² = k λ dθ* / r
• Magnetic field: dF → dB = μ₀ I ds / (4π r²) = μ₀ I r dθ / (4π r²) = μ₀ I dθ / (4π r)

Notice something? In the E and B cases, the r in ds = r dθ* cancels one power of r in the denominator. Consider this: the integrand becomes independent of r (except for the overall 1/r factor). That's why the field at the center of a semicircle scales as 1/r, not 1/r².

Center of Mass: The Classic First Encounter

A uniform wire, mass per unit length λ, bent into semicircle PQS. Day to day, total mass M = λπr. Find the center of mass.

By symmetry, x_cm = 0. Only y_cm matters.

y_cm = (1/M) ∫ y dm* = (1/λπr) ∫ (r sin θ) (λ r dθ) from 0 to π

The λ cancels. One r cancels. You get: y

The integral evaluates immediately:

[ y_{\text{cm}}=\frac{1}{\lambda\pi r}\int_{0}^{\pi}(r\sin\theta),(\lambda r,d\theta) =\frac{1}{\pi r}\int_{0}^{\pi}r^{2}\sin\theta,d\theta =\frac{r}{\pi}\Bigl[-\cos\theta\Bigr]_{0}^{\pi} =\frac{r}{\pi},(1-(-1))=\frac{2r}{\pi}. ]

Thus the center of mass of the uniform semicircular wire lies on the symmetry axis at a height

[ y_{\text{cm}}=\frac{2r}{\pi},, ]

while (x_{\text{cm}}=0) by symmetry.

Applying the same reasoning to other vector quantities

Electric field at the centre.
For a uniformly charged arc the contribution from an element (dq=\lambda,ds=\lambda r,d\theta) is

[ d\mathbf{E}= \frac{k,dq}{r^{2}};\hat{\mathbf{r}}. ]

The vector points from the element toward the centre, making an angle (\theta) with the (x)-axis. Its vertical component is (dE_{y}=dE\sin\theta). Hence

[ E_{y}= \int_{0}^{\pi}\frac{k\lambda r,d\theta}{r^{2}}\sin\theta =\frac{k\lambda}{r}\int_{0}^{\pi}\sin\theta,d\theta =\frac{k\lambda}{r},(2)=\frac{2k\lambda}{r}. ]

All horizontal components cancel, so the net field is directed along the (y)-axis with magnitude (2k\lambda/r).

Magnetic field from a current‑carrying semicircle.
A current element (d\mathbf{B}= \frac{\mu_{0}I}{4\pi}\frac{ds}{r^{2}}\hat{\boldsymbol{\phi}}) produces a contribution

[ dB = \frac{\mu_{0}I}{4\pi}\frac{r,d\theta}{r^{2}} = \frac{\mu_{0}I}{4\pi r},d\theta, ]

directed perpendicular to the plane of the arc. Integrating from (0) to (\pi) gives

[ B = \frac{\mu_{0}I}{4\pi r}\int_{0}^{\pi}d\theta = \frac{\mu_{0}I}{4r}. ]

Again, symmetry eliminates any transverse component, leaving a single scalar magnitude.

Conclusion

By choosing a convenient angular parameter and exploiting the inherent symmetry of a semicircular wire, the vertical (or out‑of‑plane) component of any vector quantity can be obtained with a single, straightforward integral. The horizontal (or in‑plane) components vanish identically, so the problem reduces to evaluating

[ \int_{0}^{\pi}! (\text{appropriate factor}),\sin\theta,d\theta, ]

or its cosine counterpart, depending on the chosen axis. This approach not only streamlines calculations for the center of mass, electric fields, magnetic fields, and the like, but also provides clear physical insight: symmetry cancels opposites, while contributions along the symmetry axis add constructively. As a result, the semicircular geometry, though simple in shape, showcases the power of symmetry‑based reasoning in evaluating line integrals across a wide range of physics problems.